Properties

Label 69696.cd
Number of curves 4
Conductor 69696
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("69696.cd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 69696.cd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
69696.cd1 69696gq4 [0, 0, 0, -3068076, -2068459184] [2] 1474560  
69696.cd2 69696gq3 [0, 0, 0, -454476, 72811024] [2] 1474560  
69696.cd3 69696gq2 [0, 0, 0, -193116, -31837520] [2, 2] 737280  
69696.cd4 69696gq1 [0, 0, 0, 2904, -1650440] [2] 368640 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 69696.cd have rank \(0\).

Modular form 69696.2.a.cd

sage: E.q_eigenform(10)
 
\( q - 2q^{5} + 4q^{7} + 6q^{13} + 6q^{17} + 8q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.